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Theorem lcvnbtwn3 29277
Description: The covers relation implies no in-betweenness. (cvnbtwn3 23302 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvnbtwn.s  |-  S  =  ( LSubSp `  W )
lcvnbtwn.c  |-  C  =  (  <oLL  `  W )
lcvnbtwn.w  |-  ( ph  ->  W  e.  X )
lcvnbtwn.r  |-  ( ph  ->  R  e.  S )
lcvnbtwn.t  |-  ( ph  ->  T  e.  S )
lcvnbtwn.u  |-  ( ph  ->  U  e.  S )
lcvnbtwn.d  |-  ( ph  ->  R C T )
lcvnbtwn3.p  |-  ( ph  ->  R  C_  U )
lcvnbtwn3.q  |-  ( ph  ->  U  C.  T )
Assertion
Ref Expression
lcvnbtwn3  |-  ( ph  ->  U  =  R )

Proof of Theorem lcvnbtwn3
StepHypRef Expression
1 lcvnbtwn3.p . 2  |-  ( ph  ->  R  C_  U )
2 lcvnbtwn3.q . 2  |-  ( ph  ->  U  C.  T )
3 lcvnbtwn.s . . . 4  |-  S  =  ( LSubSp `  W )
4 lcvnbtwn.c . . . 4  |-  C  =  (  <oLL  `  W )
5 lcvnbtwn.w . . . 4  |-  ( ph  ->  W  e.  X )
6 lcvnbtwn.r . . . 4  |-  ( ph  ->  R  e.  S )
7 lcvnbtwn.t . . . 4  |-  ( ph  ->  T  e.  S )
8 lcvnbtwn.u . . . 4  |-  ( ph  ->  U  e.  S )
9 lcvnbtwn.d . . . 4  |-  ( ph  ->  R C T )
103, 4, 5, 6, 7, 8, 9lcvnbtwn 29274 . . 3  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
11 iman 413 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  R  =  U )  <->  -.  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
12 eqcom 2368 . . . . 5  |-  ( U  =  R  <->  R  =  U )
1312imbi2i 303 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  U  =  R )  <->  ( ( R 
C_  U  /\  U  C.  T )  ->  R  =  U ) )
14 dfpss2 3348 . . . . . . 7  |-  ( R 
C.  U  <->  ( R  C_  U  /\  -.  R  =  U ) )
1514anbi1i 676 . . . . . 6  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T ) )
16 an32 773 . . . . . 6  |-  ( ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1715, 16bitri 240 . . . . 5  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1817notbii 287 . . . 4  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <->  -.  ( ( R  C_  U  /\  U  C.  T
)  /\  -.  R  =  U ) )
1911, 13, 183bitr4ri 269 . . 3  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <-> 
( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
2010, 19sylib 188 . 2  |-  ( ph  ->  ( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
211, 2, 20mp2and 660 1  |-  ( ph  ->  U  =  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    C_ wss 3238    C. wpss 3239   class class class wbr 4125   ` cfv 5358   LSubSpclss 15899    <oLL clcv 29267
This theorem is referenced by:  lsatcveq0  29281  lsatcvatlem  29298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-lcv 29268
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